PSI - Issue 44

Olivier LHERMINIER et al. / Procedia Structural Integrity 44 (2023) 528–535 LHERMINIER Olivier / Structural Integrity Procedia 00 (2022) 000–000

529

2

1. Introduction

Pushover analysis is a nonlinear static procedure for evaluating the seismic margin of build-ings accounting for their nonlinear behavior. However, a number of assumptions are made by the basic version of this method: (i) the structure has a plane of symmetry; (ii) there is a single horizontal earthquake component, parallel to the plane of symmetry; (iii) the dynamic behavior is governed by a dominant mode of vibration (with high e ff ective mass). There fore, the pushover analysis cannot be applied for the assessment of global torsion, local e ff ects or the influence of high frequency modes, and asymmetric buildings cannot be analyzed. In this paper, the Enhanced – Direct Vecto rial Approach (E-DVA) Erlicher et al. (2020) is used to take into account all the modes of irregular buildings under a multi-component earthquake. The use of a general load pattern defined by a linear combination of modal load patterns is proposed, together with an operational definition of the combination factors (called “ α -factors”) for each mode and each earthquake component; see also Erlicher and Huguet (2016); Lherminier et al. (2018). Moreover, it is shown how these factors can be used to establish a hierarchy between modes, from the one having the biggest influence on the considered structural response to the one having the smallest one. A method to determine a set of few dominant modes is given, allowing a pushover analysis which is physically significant and which remains simple. In Section 2, the notions of the response spectrum seismic analyses is presented and the new Enhanced DVA (E-DVA) load patterns are given, the α -factors associated with this load pattern are defined and a method for the definition of the set of dom inant modes is also described. In Section 3, it is applied to the non-symmetric SPEAR building under an earthquake with two horizontal components.

2. Definition of the load patterns for pushover analysis

2.1. Basic notions for pushover analyses

Let us consider a Finite Element (FE) model of a structure, with N Degrees-of-Freedom (DoFs). In a pushover analysis, this model is non-linear. The simplest method to preserve simultaneity and equilibrium is based on the application of a single load pattern on the structure. At the same time, this load pattern should permit to accurately represent the seismic structural behavior. A definition of multi-modal load pattern, for a single-component earthquake

A MC , k

A MC , k

∆ k

n = nm

n = nm + 1

(b)

(a)

(c)

MC , k ; (c) CQC

Fig. 1: Illustration of several acceleration patterns for earthquake component k = k . (a) uniform: ∆ k ; (b) CQC with n = n m modes: A

MC , k ( n = n

with n = n m modes and with residual rigid response: A

m + 1).

along direction k , is based on a quadratic modal combination formula (Figures 1b,c). Force-driven pushover analysis is characterized by the following quasi-static equilibrium equation: q pushover ( t ) = c ( t ) Q = c ( t ) M · A MC , k with A MC , k l , k = n i = 1 n j = 1 ρ i j A max , i , k l , k A max , j , k l , k and A MC , k l , k = 0 for k k (1) where q pushover ( N × 1) is the nodal force vector representing in a simplified manner the seismic action, c ( t ) is a scalar increasing pseudo-time function, Q ( N × 1) is the constant vector defining the pattern of the nodal force field, the double index l , k (or l , k ) indicates the displacement component k of the generic l − th node of the FE model, A MC , k is the vector of the positive accelerations at nodes obtained from the combination of the modal accelerations (MC: Mode Combination), the coe ffi cients ρ i j can be defined either using the CQC of Der Kiureghian (1979) and n = n m + 1 when the rigid response is taken into account or n = n m when rigid response is neglected, with n m defined as the number of modes necessary to represent the structural response when the e ff ect of the higher frequency modes ( f > f n m ) on